0:Z, →Z, A GROUP HOMOMORPHISM KER () = {0,5,10,15, 20, 25} o(13)= 6 now THAT, o" (6) FIND ?

Q: Question 1. Suppose that G = xy = yx². {e, x, x², y, yx, yx²} is a non-Abelian group with |x| = 3…

A: Given that                 G= {e,x,x2,y,yx,yx2} ba a non abelian group with o(x)=3 and o(y)=2. And…

Q: Let H = {1,5} and two operations * and on H defined as follow: %3D 15 15 1 11 1 15 5 15 5 5 Is (H,…

A: Disclaimer: Since you have asked multiple questions, we will solve the first question for you. If…

Q: I have an abstract algebra question I'm stuck on:   If G is an abelian group and n is an element of…

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Q: 6. Apply Burnside's formula to compute the number of orbits for the cyclic group G = {(1,5) o (2, 4,…

A: Solve

Q: 2. Deduce from 1 that V x Z2 is a group where V = {e, a, b, c} is the Klein-4 group. (a) Give its…

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Q: Show that the frieze group F6 is isomorphic to Z ⨁ Z2

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Q: Define ∗ on Z by x ∗ y = x + y + 4. Then: (d) Show that every element of Z is invertible with…

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Q: Show if all primitive transformations of the nonzero form x '= x ,y' = cx + dy d are a group

A: Given:  x '= x ,y' = cx + dy d

Q: Let G be a group with the property that for any x, y, z in the group,xy = zx implies y = z. Prove…

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Q: Consider the square X = [-1,1]2 = {(x, y)|x > -1, y < 1} and 0 = (0,0). Show that the fundamental…

A: image is attached

Q: Q1) Consider the group Z10X S5. Let g = (2, (345)) € Z10X S5. Find o(g). T LOV

A: as per our company guideline we are supposed to answer only one qs kindly post remaining qs in next…

Q: [Zp-(0),.] Where p is prime is an abelian group

A: We have to show that [Zp-(0),.] Where p is prime is an abelian group

Q: Let G = {[1 0 0 1] ,[−1 0 0 1] ,[1 0 0 −1], [−1 0 0 −1]} . Is G ∼= K4? If yes, give an explicit…

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Q: Q3:(A) Prove that every group of order 15 is decomposable and normal. (B) Show that (H,.) is a…

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Q: 2.3: We have showed that (aa CIR),.) is a group in our lectures Show that the map det: GL, CIR)→…

A: p

Q: be the operation on Z defined by a*b = a+b for all a,beZ. Justify the following questions. 4 Let (1)…

A: Here * be the operation on ℤ defined by a*b=a+b4 for all a, b∈ℤ. We have to justify the followings:…

Q: 1. Define x*y over R\ {-1}by x*y = x + y +xy. Prove that this structure forms an abelian group.

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Q: 1. Show that H={[0], [2], [4]} is a subgroup of a group (Z6+6). Obtain all the distinct left cosets…

A: Given that H=0,2,4 and let G=ℤ6,+6.

Q: There exists a quartic polynomial (i.e. a polynomial of degree 4) over Q whose Galois group is…

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Q: 3. In ROR under componentwise addition, let H = {(r,3r): E R}. (a) Show that H is subgroup of ROR.…

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Q: 6. Is every element of Z = {0,1,2,3, ...} has inverse with respect to addition? 7. Is (Q,.) a group?…

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Q: Question 1. Suppose that G = xy = yx². {e, x, x², y, yx, yx²} is a non-Abelian group with |æ| = 3…

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Q: LetS=R{−1} and define a binary operationon S  by a∗b=a+b+ab. Prove that (S, ∗) is an abelian group.

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Q: Prove that the group G with generators x, y, z and relations z' = z?, x² = x², y* = y? has order 1.

A: In order to solve this question we need to make the set of group G by finding x, y and z.

Q: 5. Let p be a prime. Prove that the group (x, ylx' = yP = (xy)P = 1) is infinite if p > 2, but that…

A: The solution which makes use of matrix theory is presented in detail below.

Q: 64. Express Ug(72)and U4(300)as an external direct product of cyclic groups of the form Zp

A: see my attachments

Q: 5. Prove that the group (x, y|x = yP = (xy)P = 1) is infinite if %3D %3D n> 2 but that if n = 2 it…

A: To prove that the group x, y|xp=yp=xyp=1 is infinite if p&gt;2, but that if p=2, it is a Klein…

Q: Q3:(A) Prove that every group of order 15 is decomposable and normal. (B) Show that (H,.) is a…

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Q: How do we describe all the elements in the cyclic subgroup of GL(2, R) generated by the matrix 1 ? 0…

A: For the given statement

Q: Let (G,*) be a group. Show that (G,*) is abelian iff (x * y)² = x² * y² for all x, y E G.

A: If a group G is abelian, then for any two elements x and y,    (x*y) = (y*x)   now associative…

Q: Q3: (A) Prove that 1. There is no simple group of order 200. 2. Every group of index 2 is normal.

A: Sol1:- Let G be a group of order 200  i.e O(G) = 200                = 5² × 8. G contains k Sylows…

Q: 5. Consider the group (R+, 0). Prove that the function F: R -R given by: F (x.y) = (x +y.r-y) is a…

A: F: R2 → R2F (x, y) = (x +y, x-y)Let (x1, y1) ,  (x2, y2) = R2 (x1, y1) +  (x2, y2) =  (x1+ x2, y1+…

Q: 2) Prove that Zm × Zn is a cyclic group if and only if gcd(m, n) cyclic group Z; x Z4. = 1. Find all…

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Q: Let S = R\{-1}. Define * on S by a * b = a+b+ ab. Prove that (S, *) is an abelian group.

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Q: (d) A cyclic group of order n has no proper nontrivial subgroup if and only if n is prime. (e) If o…

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Q: Show that Z12 is not isomorphic to Z2 ⊕ Z6.  ℤn denotes the abelian cyclic group of order n. Justify…

A: To show : ℤ12 is not isomorphic to ℤ2⊕ℤ6 Pre-requisite : P1. A group G is said to be cyclic if there…

Q: How do we describe all the elements in the cyclic subgroup of GL(2, R) generated by the matrix 1 ? 1…

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Q: 5. Prove that the cyclic group Z/15Z is isomorphic to the product group Z/3Z × Z/5Z.

A: Definitions: Isomorphism: A mapping between two sets is called an isomorphism if it is one-to-one,…

Q: If G is a non-abelian group of order 8 with Z(G) {e}, prove that |Z(G)| = 2.

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Q: Suppose that f:G - G such that f(x) = axa". Then fis a group homomorphism if and only if O a^3 = e a…

A: f:G→G such that  fx=axa2 We know that f is a homomorphism if fxy=fxfy for all x, y∈G

Q: 2. Are the groups Z/2Z x Z/12Z and Z/4Z x Z/6Z isomorphic? Why or why not?

A: Here we have to show that given groups are isomorphic

Q: 2- Let (C,) be the group of non-zero -complex number and let H = {1, –1, i, -1}. Show that (H,;) is…

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Q: 4) If center of G equal G (ie cent(G) = G) then G is not necessary abelian group. true O false

A: Note: Since you have asked multiple question, we will solve the first question for you. If you want…

Q: 22. Prove that the set = {(₁ ~ ) 1} x) | : x, y ≤ R, x² + y² = 1 = SO(2) = forms an abelian group…

A: Given: 22. SO(2)=x-yyx : x, y∈ℝ, x2+y2=1 To show: The given set is a group with respect to…

Q: a. Show that (Q\{0}, + ) is an abelian (commutative) group where is defined as a•b= ab b. Find all…

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Q: 7. You have previously proved that the intersection of two subgroups of a group G is always a sub-…

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Q: Q2.6 Question 1f How many Abelian groups (up to isomorphism) are there of order 36? O 2 O 3 O 4

A: Option D is correct answer

Q: 1. Consider the groups (R+, ) and (R,+). Then R* and R are isomorphic under the mapping $(x) = log10…

A: We use the definition of cosets, isomorphisms to answer these questions. The detailed answer well…

Q: Show if all primitive transformations of the nonzero form x '= x ,y' = cx + dy d are a group.

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Q: (4) Find the Galois group of the polynomial r + 1.

A: Since you have asked multiple question, we will solve any one question for you. If you want any…

Unitary Method

The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products. 

Speed, Time, and Distance

Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.

Profit and Loss

The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.

Units and Measurements

Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.